A Convergent Hybridizable Discontinuous Galerkin Method for Einstein--Scalar Equations

TL;DR

This paper introduces and analyzes a hybridizable discontinuous Galerkin method for solving the Einstein--scalar equations in spherical symmetry, demonstrating stability, convergence, and applicability to complex dynamics.

Contribution

The paper develops a novel HDG scheme for Einstein--scalar equations, proving stability, error bounds, and showcasing its effectiveness through numerical experiments.

Findings

Proved local well-posedness of the scheme.

Established optimal order L2 error bounds for the main variable.

Numerical results confirm convergence and capture complex dynamics.

Abstract

We propose and analyze a hybridized discontinuous Galerkin (HDG) method for the spherically symmetric Einstein--scalar system in Bondi gauge. After rewriting the model as a local first-order PDE--ODE system by introducing suitable scaled variables, we construct a semidiscrete scheme in which the element unknowns are computed locally and the coupling is carried by traces on the mesh skeleton. In the present radial setting, these traces can be eliminated recursively, so that only the main evolution variable is advanced in time, while the metric variables are recovered from discrete constraint relations. We prove local semidiscrete well-posedness, derive a global \(L^2\)--stability estimate, establish an optimal order \(L^2\) error bound for the main evolution variable for polynomial degree \(k\ge 1\), and obtain reconstruction error estimates for the metric variables and the associated mass functional. Numerical experiments verify the predicted spatial convergence rate and illustrate qualitative features of the Einstein--scalar dynamics, including large-data collapse profiles and smooth-pulse evolution.